This is another project involving one of Euler's favorite topics : the finding of curves satisfying maximum or minimum properties under various circumstances ; in the first chapter, a general method of finding such curves is set out and numerous examples of its use presented. In the second and third chapters, the work examines increasingly complicated applications of what was later called the Calculus of Variations. At first determined functions are treated, in which the values of the derivatives of the curve are known at any point; later more abstract constructions are introduced in chapters 2 & 3 where Euler deals with functionals related by a general integral formula; for example, the theory developed can be used to find the shape of the curve a body can fall along in the minimum time subject to various forms of resistance, etc. Chapter 4, section 7 sets out the precepts for the 5 kinds of extrema considered ; this is a very useful summary.
It is of course a fundamental work in the establishing the Calculus of Variations by Euler and Lagrange a little later, being a sort of hybrid of ordinary integration and the solving of differential equations satisfying certain conditions ; the functions being integrated are usually not algebraic or transcending, but involve differential formulas that may be expanded out in terms of the usual x, y , p, q, r , etc. , and so special methods are needed to solve such problems. A good place to look initially for a modern appraisal of the work is the article by Craig G. Fraser, 'The Origins of Euler's Variational Calculus' in the Archives of the Exact Sciences. Vol. 47, No. 2 (June 1994), pp. 103-141. Pub. by Springer. However, Chapter 2 of Herman H. Goldstine's classic work ' A History of the Calculus of Variations, from the 17th through the 19th Century', pub. by Springer in 1980, must take pride of place, and should be consulted regularly with this translation, although of course in one chapter only a glimpse of the contents of Euler's book can be set out. Other places to look include of course Wikipedia ; the book by Routh on Rigid Body Dynamics includes a chapter on the Vis Viva which is useful for Addition II ; and of course Salmon' s classic work on columns is useful for Addition I.
Click here for the 1st Chapter : Concerning the Method of finding the maxima and minima of curved lines generally.
This is an introductory chapter in which most of what lies ahead in the following chapters is surveyed perhaps in a rather abstract view. Thus, Euler distinguishes between the 'ordinary' max. and min. of a given curve, and the present situation, where the curve itself is to be varied in a given domain to produce an extreme value. A great deal of careful argument is gone into, the relevance of which may only become apparent on reading further chapters. The use of an integral Zdx as the formula carrying the formula was revolutionary at the time, as was the use of x, y, p, q, etc. as variables; a most convenient notation, not involving derivatives directly greater than first order. It is a good idea to return to this introductory chapter and to refresh oneself on the ideas presented. See H. Goldstine p.67 onwards for a review of this chapter and the next.
Click here for Chapter 2a : A Method for finding the absolute maxima and minima for curved lines.
This is a long chapter containing much material, which I have split into two parts for computing convenience. Here Euler has been able to set out the conditions for the max. or min. of a curve contained within an integral in a straight forwards manner, and depending on y, p, q, etc.. The basic idea consists of extending some arbitrary y coordinate of the max. or min. curve by the infinitesimal amount nv, calculating the changes arising in dZ at neighbouring points and equating these changes to zero, so allowing Euler to set up a necessary condition for a max. or min. in terms of simple differential equations, depending on the form of dZ , and into an increasing number of terms. Numerous examples are given, the work is continued in part 2b. At present determinate formulas only are considered.
Click here for Chapter 2b : A Method for finding the absolute maxima and minima for curved lines.
Click here for Chapter 3 :On finding max./min. …. with indeterminate magnitudes present. This is a long chapter in which the mathematical procedures needed to resolve various physical problems are set out ; such problems involve bodies sliding down unknown curves according to some law of resistance, with the aim of minimizing the time of falling, etc. Thus a number of unknown functions are present in the integrand to be minimized. This leads Euler to an extension of his original scheme, so that the calculations become more involved, but still follow the same general lines. See H. Goldstine p.73 onwards for a review of this chapter.
Click here for Chapter 4 :Concerning the use of the method now treated in the resolution of various kinds of questions. This is another long chapter ; however, in section 7 Euler has summarized the 5 main methods he has set up in chapter 3 for finding optimal curves under various conditions ; the examples help make the work of chapter 3 much easier to understand; one wonders why he took so long to introduce his examples….. See H. Goldstine p.84 onwards for a review of this chapter.
Click here for Chapter 5 : A method of finding that curve among all the curves with a given property, which may be endowed with the property of being a max. or min.
This is another long chapter ; here Euler considers the introduction of an extra condition to be satisfied by the max. or min. curve, and satisfied by all the curves : i.e. an isoperimetric condition ; this involves the use of extending two neighbouring applied lines by independent incremental amounts in defining the common property, as well as the above single applied line, or y-coordinate. See H. Goldstine p.93 onwards for a review of this chapter. Many examples are given in a most thorough investigation.
Click here for Chapter 6 : A method for determining that curve among the curves endowed with several common properties, which may be provided with the max. or min. property.
In this final chapter, before moving on to appendices, Euler examines the problem of variation when some number of common conditions are present, and the max. min. value of a curve is to be found satisfying these conditions. The third proposition lands Euler in trouble when he tries to generalize and to produce the same result extended by his new line of reasoning, that he found before from the general method introduced. Nevertheless, the old method works, and is used to solve three more examples.
Click here for Appendix 1A : The curves associated with elastic laminas.
In this appendix, later added to the main work, Euler sets out to show that his method of finding maxima or minima curves associated with generalized functions in the form of integrals, can be applied to finding the shape of loaded laminas or ribbons, as had then recently been established in a straightforward method from mechanics by Daniel Bernoulli, following on the earlier work of his uncle, James Bernoulli. Most of the first part of this appendix, so subdivided for convenience, is given over to finding the nine classes or kinds of shapes adopted by a flexed lamina under different end conditions. An English translation exists already in Isis (1933) by Oldfather et al., of which I have just become aware, and have not referred to here.
Click here for Appendix 1B : The curves associated with elastic laminas cont'd.
This completes the deflections and vibrations of beams as presented by Euler at this time; he was to correct an error spotted by Daniel Bernoulli, which we will deal with here soon.
Click here for Appendix II : The motion of projectiles in a non-resisting medium, determined by the method of maxima and minima .
This completes Euler's work at this time; in this last section is the germ of the calculus of variations; Euler is able to derive the equations of motion of projectiles and orbits according to the vis viva principle ; this is very close to the correct theory, though he knows his effort is not complete, due to the state of physics at the time.
Click here for E296 : The elements of the calculus of variations.
Soon after Lagrange sent Euler a letter and his method for shortening Euler's calculations in this treatise, Euler produced this paper in which he sets out Lagrange's contributions in a clear manner.
Click here for E297 : The analytical explanation of the method of maxima and minima.
In this second paper, Euler revisits his formulas and rewrites his main results in term of variations; this is probably the best place to look if you do not have the desire or the time to read the above book. The most satisfactory thing of course is to do both.
Ian Bruce. Aug. 8th , 2013 latest revision. Copyright : I reserve the right to publish this translated work in book form. You are not given permission to sell all or part of this translation as an e-book. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. See note on the index page.Please feel free to contact me if you wish by clicking on my name here, especially if you have any relevant comments or concerns.